Cubic equation solution

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

A variety of equations-of-state have been applied to supercritical fluids, ranging from simple cubic equations like the Peng-Robinson equation-of-state to the Statistical Associating Fluid Theoiy. All are able to model nonpolar systems fairly successfully, but most are increasingly chaUenged as the polarity of the components increases. The key is to calculate the solute-fluid molecular interaction parameter from the pure-component properties. Often the standard approach (i.e. corresponding states based on critical properties) is of limited accuracy due to the vastly different critical temperatures of the solutes (if known) and the solvents other properties of the solute... 

Equation 13-39 is a cubic equation in terms of the larger aspect ratio R2. It can be solved by a numerical method, using the Newton-Raphson method (Appendix D) with a suitable guess value for R2. Alternatively, a trigonometric solution may be used. The algorithm for computing R2 with the trigonometric solution is as follows ... 

For instance, a quadratic expression for C (T) will require the solution of a cubic equation in T. ... 

This is a cubic equation in the compressibility factor Z, which can be solved analytically. The solution of this cubic equation may yield either one or three real roots4. To obtain the roots, the following two values are first calculated4 ... 

Solutions for cAlcAo from equation 4.3-9 are given in the third column in Table 4.1. For n = 3/2 and 3, the result is a cubic equation in (cAlcAo)m and cAlcAo,respectively. The analytical solutions for these are cumbersome expressions, and the equations can be solved numerically to obtain the curves in Figure 4.4. 

Plotting the parabola (broken curve) on the left of (1.40) with the rectangular hyperbola (solid curve) on the right (Fig. 1.4) shows that this cubic equation in eJlk has three roots given by the intersections pi, P2 and p3. However, since eJlk 1 at pi, this root is rejected, so (1.40) has two real solutions, P2 and p3, whose corresponding /rfc-values give rise to two P-states. 

Sigma (a) bonds Sigma bonds have the orbital overlap on a line drawn between the two nuclei, simple cubic unit cell The simple cubic unit cell has particles located at the corners of a simple cube, single displacement (replacement) reactions Single displacement reactions are reactions in which atoms of an element replace the atoms of another element in a compound, solid A solid is a state of matter that has both a definite shape and a definite volume, solubility product constant (/ p) The solubility product constant is the equilibrium constant associated with sparingly soluble salts and is the product of the ionic concentrations, each one raised to the power of the coefficient in the balanced chemical equation, solute The solute is the component of the solution that is there in smallest amount, solution A solution is defined as a homogeneous mixture composed of solvent and one or more solutes. 

Notice that KACSYKA has obtained the roots analytically and that numeric approximations have not been made. This demonstrates a fundamental difference between a Computer Algebra system and an ordinary numeric equation solver, namely the ability to obtain a solution without approximations. 1 could have given KACSYKA a "numeric" cubic equation in X by specifying numeric values for A and B. KACSYKA then would have solved the equation and given the numeric roots approximately or exactly depending upon the specified command. 

There are three real solutions to this cubic equation (why all the solutions are real in this case for which the M matrix is real and symmetric will be made clear later) ... 

The roots of quadratic and cubic equations are well known as algebraic expressions of the equation s coefficients, and hence this section is comletely disconnected from the rest of the chapter. Nevertheless, these simple problems are so frequently encountered that we cannot ignore their special solutions. 

For the hysteresis limit we require, as usual, that F = Fx = Fxx = 0. Two equalities give x and ires. The third then leads to eqn (7.34) relating gad to rN. The isola condition F = Fx = Ft = 0 is best handled parametrically, as x cannot be eliminated so readily (it is given by the solution of a cubic equation in terms of gad). The parametric forms have been given as eqns (7.35) and (7.36). 

The stationary states are given by the solutions of a cubic equation ... 

In practice, the solution of polynomial equations is problematic if no simple roots are found by trial and error. In such circumstances the graphical method may be used or, in the cases of a quadratic or cubic equation, there exist algebraic formulae for determining the roots. Alternatively, computer algebra software (such as Maple or Mathematica, for example) can be used to solve such equations... 

As stressed in Section II, the coefficients y, M, are to be determined from suitable solvability conditions. One finds that to order e2 the solvability conditions yield y, = 0. Thus, the amplitudes p, p2 cannot be determined to this order. To order e3 one obtains a nontrivial result, in the form of two coupled cubic equations for p, and p2. Among the possible solutions of these equations one obtains rotating wave solutions. Setting p2 = 0 one finds a clockwise wave ... 

Attempts to simplify the equation produce a cubic equation in y, giving no straightforward means to a numerical solution. You can, however, easily obtain a numerical solution for y with a hand calculator. Start by solving for y in terms of y2 as follows... 

The completely reliable computational technique that we have developed is based on interval analysis. The interval Newton/generalized bisection technique can guarantee the identification of a global optimum of a nonlinear objective function, or can identify all solutions to a set of nonlinear equations. Since the phase equilibrium problem (i.e., particularly the phase stability problem) can be formulated in either fashion, we can guarantee the correct solution to the high-pressure flash calculation. A detailed description of the interval Newton/generalized bisection technique and its application to thermodynamic systems described by cubic equations of state can be found... 

Accordingly, the solution of (74) was found from a quadratic equation. It is clear that the possibilities to produce exact explicit formulas along this line are limited the solutions for n = 4 and n = 5 have been accomplished [17, 18] by means of the relevant cubic equations, but the answer for larger n values have not been found previously. 

The solution for a general cubic equation (analogous to Equation 1.5) is vastly more tedious, and never used in practice. This problem can be solved by assuming that the small amount of dissolved PbCh does not substantially change the chloride concentration (x < 0.1), which gives... 

Experimental results are presented for high pressure phase equilibria in the binary systems carbon dioxide - acetone and carbon dioxide - ethanol and the ternary system carbon dioxide - acetone - water at 313 and 333 K and pressures between 20 and 150 bar. A high pressure optical cell with external recirculation and sampling of all phases was used for the experimental measurements. The ternary system exhibits an extensive three-phase equilibrium region with an upper and lower critical solution pressure at both temperatures. A modified cubic equation of a state with a non-quadratic mixing rule was successfully used to model the experimental data. The phase equilibrium behavior of the system is favorable for extraction of acetone from dilute aqueous solutions using supercritical carbon dioxide. 

The last term, the fluid s isothermal compressibility, can be reasonably predicted from a two-parameter, cubic equation of state (EOS) such as the Redlich-Kwong EOS or the Peng-Robinson EOS.(14,15) The fluid s isothermal compressibility was determined using the Redlich-Kwong EOS to evaluate the derivative ( Vmmob/3P) in eq. 12 due to the ease of finding an analytical solution,... 

The second term on the RHS of eq. 22 is the volume expansivity which can be calculated from a two-parameter, cubic equation of state such as the Redlich-Kwong EOS.(14) We chose the Redlich-Kwong EOS because the "a" term is independent of temperature, which simplifies the procedure of solving for the analytical solution. Using the method of implicit differentiation with the Redlich-Kwong equation of state,... 

Some equations, such as that of van der Waals, are trivial to solve for some variables (e.g., P or T), but more difficult to solve others (V). As discussed in Chapter 1, the van der Waals equation may be rearranged into a cubic equation for V, and formulas are available to obtain the roots for such equations. Our objective here is to discuss more general methods for obtaining approximate solutions for such equations. One way to approach the solution of the general equation,... 

---